The OPERA neutrino experiment is designed to perform the first observation of neutrino oscillations in direct appearance mode in the $\nu_\mu \to \nu_\tau$ channel, via the detection of the $\tau$-leptons created in charged current $\nu_\tau$ interactions. The detector, located in the underground Gran Sasso Laboratory, consists of an emulsion/lead target with an average mass of about 1.2 kt, complemented by electronic detectors. It is exposed to the CERN Neutrinos to Gran Sasso beam, with a baseline of 730 km and a mean energy of 17 GeV. The observation of the first $\nu_\tau$ candidate event and the analysis of the 2008-2009 neutrino sample have been reported in previous publications. This work describes substantial improvements in the analysis and in the evaluation of the detection efficiencies and backgrounds using new simulation tools. The analysis is extended to a sub-sample of 2010 and 2011 data, resulting from an electronic detector-based pre-selection, in which an additional $\nu_\tau$ candidate has been observed. The significance of the two events in terms of a $\nu_\mu \to \nu_\tau$ oscillation signal is of 2.40 $\sigma$.
